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Homework Help: Electric Fields In Matter

  1. May 26, 2008 #1
    1. The problem statement, all variables and given/known data
    A thick, spherical shell (inner radius a, outer radius b) is made of dielectric material with a 'frozen in' polarization
    [tex]\textbf{P} = \frac{k}{r} \textbf{\hat r} [/tex]
    where k is a constant and r the distance from the centre. (There is no free charge). Find the electric field in each o the three regions by two methods:

    a) Locate all the bound charge, nd use Gauss' Law to calculate the field it produces
    b) Use [tex]\oint \textbf{D}.d\textbf{a} = Q_{f enc} [/tex] to find [tex]\textbf{D} [/tex] and then find the electric field

    3. The attempt at a solution

    a) bound charge density is [tex]\rho _b -\nabla \textbf{P} = -frac{k}{r^2} [/tex]

    Gauss' Law: [tex] \oint E.da = Q_{enc}/\epsilon _0 [/tex]
    In the dielectric material:
    [tex]Q_{enc} = \int \rho _b = -4\pi \int _a^r dr '= 4\pi k(a-r)[/tex]
    I integrated from a to r because that's the only region with charge.
    From Guass' Law:

    [tex]E4\pi r^2 = 4\pi k(a-r)/\epsilon_0 [/tex]

    [tex]E = k(a-r)/(\epsilon_0 r^2)[/tex]

    For r<a the electric field is zero because there is no bound charge. Similarly, outside the sphere
    [tex]E = k(a-b)/(\epsilon_0 r^2)[/tex]

    [tex]\oint \textbf{D}.d\textbf{a} = Q_{f enc} = 0 [/tex]
    because there is no free charge. So
    [tex] \textbf{D} = \epsilon _0 \textbf{E} + \textbf{P} = 0 [/tex]

    [tex] \textbf{E} = -\textbf{P}/\epsilon_0 [/tex]

    Where did I go wrong here?
  2. jcsd
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